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Chapter 5: Problem 57

Use the sum-to-product formulas to find the exact value of the expression. $$\sin 75^{\circ}+\sin 15^{\circ}$$

Short Answer

Expert verified
The exact value of the expression \(\sin 75^\circ + \sin 15^\circ\) is given by \(\sqrt{6}/2\)

Step by step solution

01

Identify Appropriate Sum-to-Product formula

The appropriate version of the Sum-to-Product formula to use in this exercise is: \(\sin a + \sin b = 2\sin\left(\frac{a+b}{2}\right)\cos\left(\frac{a-b}{2}\right)\). This formula will aid in simplifying the given expression.
02

Substitute into Sum-to-Product formula

The given values are substituted into the formula, letting \(a = 75^\circ\) and \(b = 15^\circ\). Thus, the formula becomes: \(\sin 75^\circ + \sin 15^\circ = 2\sin\left(\frac{75+15}{2}\right)\cos\left(\frac{75-15}{2}\right)\)
03

Calculate values in formula

Calculate the bracketed values. The Sin and Cos values can be found on the trigonometric table. \(\frac{75+15}{2} = 45^\circ\) and \(\frac{75-15}{2} = 30^\circ\). Therefore, the revised formula is: \(2\sin 45^\circ \cos 30^\circ\)
04

Use trigonometric table values to compute the result

From standard trigonometric values, we know that \(\sin 45^\circ = \sqrt{2}/2\) and \(\cos 30^\circ = \sqrt{3}/2\). When these are substituted in our previous step, the expression becomes \(2 * \sqrt{2}/2 * \sqrt{3}/2 = \sqrt{6}/2\)

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